Explicit numerical computation of normal forms for Poincaré maps

We present a methodology for computing normal forms in discrete systems, such as those described by Poincaré maps. Our approach begins by calculating high-order derivatives of the flow with respect to initial conditions and parameters, obtained via jet transport, and then applying appropriate projections to the Poincaré section to derive the power expansion of the map. In the second step, we perform coordinate transformations to simplify the local power expansion around a dynamical object, retaining only the resonant terms. The resulting normal form provides a local description of the dynamics around the object, and shows its dependence on parameters. Notably, this method does not assume any specific structure of the system besides sufficient regularity.

To illustrate its effectiveness, we first examine the well-known Hénon–Heiles system. By fixing an energy level and using a spatial Poincaré section, the system is represented by a 2D Poincaré map. Focusing on an elliptic fixed point of this map, we compute a high-order normal form, which is a twist map obtained explicitly. This means that we have computed the invariant tori inside the energy level of the Poincaré section. Furthermore, we explore how both the fixed point and the normal form depend on the energy level of the Poincaré section, deriving the coefficients of the twist map as a power series of the energy level. This approach also enables us to obtain invariant tori inside nearby energy levels. We also discuss how to obtain the frequencies of the torus for the flow. We include a second example involving an elliptic periodic orbit of the spatial Restricted Three-Body Problem. In this case the map is 4D, and the normal form is a multidimensional twist map.